The Social Climbing Game

Abstract

The structure of societies depends, to some extent, on the incentives of the individuals they are composed of. We study a stylized model of this interplay, that suggests that the more individuals aim at climbing the social hierarchy, the more society’s hierarchy gets strong. Such a dependence is sharp, in the sense that a persistent hierarchical order emerges abruptly when the preference for social status gets larger than a threshold. This phase transition has its origin in the fact that the presence of a well defined hierarchy allows agents to climb it, thus reinforcing it, whereas in a “disordered” society it is harder for agents to find out whom they should connect to in order to become more central. Interestingly, a social order emerges when agents strive harder to climb society and it results in a state of reduced social mobility, as a consequence of ergodicity breaking, where climbing is more difficult.

Appendix: Ergodicity of the Dynamics

Let ΓC(N,M) be the space of connected graphs with N vertices and M edges. In order to prove ergodicity, we have to show that, with a finite number of basic moves, we can reach any connected graph in ΓC(N,M), starting from another arbitrary graph in the same set. Before delving into the technical details, we give a simple intuitive sketch of this proof.

For a finite value of β, the dynamics consists of reversible moves as the one depicted in Fig. 9; such moves can be thought of as a “sliding” of the edge eik on the path of length one (k,j) from vertex k to vertex j. The key observation to prove the ergodicity by induction is that, since the graph is finite and connected, there always exists a path of minimum length that connects two arbitrarily chosen vertices in the graph. Then, we can proceed in three steps.

1.

Let there be two graphs in ΓC(N,M) which differ from each other only by an edge incident on the same vertex, vk. We first prove that by means of basic moves, we can transform one into the other. To do so, it suffices to slide the edge along the path that connects the other end of the edge, which we know to exist because the graph is connected (Proposition 1, Proposition 2).

2.

Let there be two graphs in ΓC(N,M) that differ by an edge with arbitrary ends. By applying the previous step twice, we show that there exists a finite set of moves that allows us to reach one configuration starting from the other (Proposition 3).

3.

Finally, let there be two arbitrary graphs in ΓC(N,M). Moving one edge at time, we show by induction that it is possible to reach one graph starting from the other with a finite number of moves. Thus, the ergodicity is proved (Proposition 4).

Definition 1

(c-swap)

Let us choose a labeling for the space of vertices V={v1,…,vN} and an induced labeling for the edges E={eij} where eij=eji=(vi,vj) denotes the undirected edge between vi and vj. Let us define a transformation \(\sigma_{ij}^{ik}: \varGamma^{C}(N,M) \mapsto\varGamma^{C}(N,M) \), called corner swaps (c-swaps), as following

Proposition 1

Let\(\mathcal{G}=(V,E)\)and\(\mathcal{G}'=(V,E')\)be two graphs inΓC(N,M) that differ by an edge incident on the same vertex, i.e. |E|=|E′|=M, |E′∩E|=M−1, E∖E′={eik} andE′∖E={eij}, and such that the shortest pathPfromvktovjdoes not contain neithervinor any of its neighbors.

There exists an integerland a finite sequence of graphs inΓC(N,M), \(\mathcal{G}^{n}\)such that:

Proof

Let l be the length of P.

Let \(v_{k_{1}}\) be the unique neighbor of vk that lies in P. If we set \(\mathcal{G}^{1}=\sigma_{i k_{1}}^{i k} ( \mathcal{G})\), the c-swap reduces the distance between vi and vj, since the neighbor of vk that lies in P must have a distance l−1 from vj. We reiterate the procedure on \(\mathcal{G}^{1}\) and obtain in such a way a sequence of graphs that satisfies property (ii). Now, since at any step the length of P diminishes by 1, after the l-th step, in the graph \(\mathcal{G}^{l}\)vi and vj will be neighbors. Thus, since no other edge was changed by applying c-swaps, \(\mathcal {G}^{l}=\mathcal{G}'\) proving property (i). □

Proposition 2

Let\(\mathcal{G}=(V,E)\)and\(\mathcal{G}'=(V,E')\)be two graphs inΓC(N,M) which differ by an edge incident on the same vertex, i.e. |E|=|E′|=M, |E′∩E|=M−1, E∖E′={eik} andE′∖E={eij}.

There exist an integerland a finite sequence of graphs inΓC(N,M), \(\mathcal{G}^{n}\)such that:

Proof

Let P be the shortest path in \(\mathcal{G}\) from vk to vj that does not contain (vk,vi).

There are four possible cases:

(i)

P does not contain neither vi nor any of its neighbors other than vk. The thesis is proven applying Proposition 1 directly to P.

(ii)

P contains vi. Let P1 be the shortest path from vk to vi that does not contain the edge (vk,vi). Let P2 be the shortest path from vi to vj, clearly P=P1⊕P2, where ⊕ is the path concatenation. Since by construction there are no neighbors of vk in P2 (otherwise P would not contain vi) we can apply Proposition 1 and reach \(\mathcal {G}''=(V,(E\setminus \{e_{ki}\})\cup\{e_{kj}\})\); on the other hand there cannot be neighbors of vj in P1 (otherwise there would be a shortest path not containing vi) and thus applying again Proposition 1 along P1 we reach \(\mathcal{G}'\) proving the thesis.

(iii)

P does not contain vi but two of its neighbors, c and f such that c≠vk, f≠vk and |c,vk|<|f,vk|, where |⋅,⋅| represents the graph distance between two vertices. We first note that c and f must be neighbors, otherwise P should include vi. Then, as in case (ii), by minimality we can write P=P1⊕(c,f)⊕P2 where P1 is the shortest path from vk to c and P2 is the shortest path from f to vj. It is easy to see that Q2=(vi,f)⊕P2 is a shortest path from vi to vj: if it were not so, there would exist a path \(Q_{2}'\) from vi to vj strictly shorter than Q2, but in that case \(P_{1}\oplus(c,v_{i})\oplus Q_{2}'\) would be a shortest path from vk to vj containing vi, in contradiction with our hypotheses. A similar argument holds for Q1. As before, since, by minimality, there cannot be neighbors of vk in P2, it is possible to reach the graph \(\mathcal{G}''=(V,(E\setminus\{e_{ki}\})\cup\{e_{kj}\})\) by applying Proposition 1 to Q2; since by minimality there cannot be neighbors of vj in P1, we can apply Proposition 1 to \(\mathcal{G}''\) along Q2 and reach \(\mathcal{G}'\) proving the thesis.

(iv)

The shortest path P contains only one neighbor of vi other than vk, let us call it m. As before, P=P1⊕P2 where P1 is the shortest path from vk to m and P2 is the shortest path from m to vj. Since by construction there cannot be other neighbors of i in P2, we can apply Proposition 1 to P2 and reach the graph \(\mathcal{G}^{*}=(V,(E\setminus\{e_{im}\})\cup\{e_{ij}\})\). On the other hand, by construction there cannot be neighbors of vi in P1 other than vk and thus we can apply Proposition 1 to P2 and reach \(\mathcal{G}'\) proving the thesis.

□

Proposition 3

Let\(\mathcal{G}=(V,E)\)and\(\mathcal{G}'=(V,E')\)be two graphs inΓC(N,M) such that |E|=|E′|=Mand |E∩E′|=M−1. Let us assume that, in particular, E={eij}∪(E∩E′) andE′={ehk}∪(E∩E′).

Thus there exists an integerland a finite sequence of graphs inΓC(N,M), \(\mathcal{G}^{n}\)such that:

Proof

Let us define the graph \(\mathcal{G}''=(V,E'')\) such that E″=(E∖{eij})∪{eih}. Applying Proposition 2 first to graphs \(\mathcal{G}\) and \(\mathcal{G}''\) and then to graph \(\mathcal{G}''\) and \(\mathcal{G}'\) proves the thesis. □

Definition 2

(g-swap)

Let \(\mathcal{G}=(V,E)\) and \(\mathcal{G}'=(V,E')\) be two graphs in ΓC(N,M) which differ at most by an edge, that is such that |E|=|E′|=M and |E∩E′|=M−1. Let us assume that, in particular, E={eij}∪(E∩E′) and E′={ehk}∪(E∩E′).

We define a global swap org-swap of the edge eij to the edge ehk a transformation such that:

$$ \mathcal{G}'=\varSigma_{ij}^{hk}(\mathcal{G}). $$

(10)

Proposition 3 simply states that any global swap can be obtained as the composition of a minimal set of corner swaps between adjacent vertices.

Proposition 4

Let\(\mathcal{G}=(V,E)\)and\(\mathcal{G}=(V,E')\)be two graphs inΓC(N,M). There exists an integerdand a sequence of graphs\(\mathcal{G}^{n}(V, E_{n})\)inΓC(N,M) such that:

(i)

\(\mathcal{G}=\mathcal{G}^{0}\)and\(\mathcal{G}'=\mathcal{G}^{d}\).

(ii)

For all 0≤n<dthere exist four verticesvi, vj,vhandvksuch that\(\mathcal{G}^{n+1}=\varSigma _{ij}^{hk}(\mathcal{G}^{n})\).

Proof

Let \(\mathcal{Z}=(V,Z=E\cap E')\), and let us define δ=|Z|. We proceed by induction on the number δ.

Base case

If δ=M−1, the Thesis is trivially true because of Proposition 3.

Inductive step

Let us assume that the Thesis holds for δ=M−d, we want to show that this implies that it also holds for δ=M−d−1, with d<M−1. Let us assume that \(\mathcal {G}= (V,E)\) and \(\mathcal{G}' = (V,E')\) are such that |E′∩E|=M−d−1. Let eij∈E∖(E∩E′) and ehk∈E′∖(E∩E′). Moreover, let E″=(E∖{eij})∪{ehk}. By construction, |E∩E″|=M−1 and |E′∩E″|=M−d. Finally, let \(\mathcal{G}''= (V,E'')\). Since \(\mathcal{G}''\) and \(\mathcal{G}'\) differ by M−d edges, by inductive assumption there exists a sequence \(\mathcal{G}^{i}\), with i∈[0,d], such that \(\mathcal{G}^{0} = \mathcal{G}'\) and \(\mathcal{G}^{d} = \mathcal{G}''\), that satisfies the Thesis. Moreover, by Proposition 3, there exists a g-swap such that \(\mathcal{G}= \varSigma _{hk}^{ij}(\mathcal{G}'')\). Thus, the complete sequence \(\mathcal {G}'=\mathcal{G}^{0},\mathcal{G}^{1},\ldots,\mathcal{G}''=\mathcal {G}^{d},\mathcal{G}=\mathcal{G}^{d+1}\) satisfies the Thesis.

□

Proposition 4 and Proposition 3 state simply that any two connected graphs with the same number of edges can be obtained one from the other applying a finite sequence of c-swaps. Moreover, since the number of edges is finite, then there must be a minimal sequence of c-swaps that connects any two of such graphs. Since, for finite β, all c-swaps are allowed with nonzero probability, this proves the ergodicity.